Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems (Communications in Computational Physics, 2018)
We study the construction of symplectic Runge-Kutta methods for stochasticHamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise, respectively, are consideredin this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated,and the corresponding conditions for SRK methods to preserve the symplecticproperty are given. Based on the weak/strong order and symplectic conditions, someeffective schemes are derived. In particular, using the algebraic computation, we obtainedtwo classes of high weak order symplectic Runge-Kutta methods for SHS witha single multiplicative noise, and two classes of high strong order symplectic RungeKuttamethods for SHS with multiple multiplicative and additive noise, respectively.The numerical case studies confirm that the symplectic methods are efficient computationaltools for long-term simulations.